Transcript Bayes for Beginners

Bayes for Beginners
Presenters: Shuman ji & Nick Todd
Statistic Formulations.
P(A): probability of event A occurring
P(A|B): probability of A occurring given B occurred
P(B|A): probability of B occurring given A occurred
P(A,B): probability of A and B occurring simultaneously
(joint probability of A and B)
Joint probability of A and B
P(A,B) = P(A|B)*P(B) = P(B|A)*P(A)
Bayes Rule
• True Bayesians actually consider conditional probabilities as more basic
than joint probabilities . It is easy to define P(A|B) without reference to
the joint probability P(A,B). To see this note that we can rearrange the
conditional probability formula to get:
• P(A|B) P(B) = P(A,B)
by symmetry:
• P(B|A) P(A) = P(A,B)
• It follows that:
• which is the so-called Bayes Rule.
• Thus, Bayes Rule is a simple mathematical formula used for
calculating conditional probabilities
Bayesian Reasoning
ASSUMPTIONS
P(A) =1% of women aged forty who participate in a
routine screening have breast cancer
P(B|A)=80% of women with breast cancer will get
positive tests
9.6% of women without breast cancer will also get
positive tests
EVIDENCE
A woman in this age group had a positive test in a routine
screening
PROBLEM
What’s the probability that she has breast cancer?
= proportion of cancer patients with positive results,
within the group of All patients with positive results
--- P(A|B)
P(B)= proportion of all patients with positive results
Bayesian Reasoning
ASSUMPTIONS
100 out of 10,000 women aged forty who
participate in a routine screening have breast
cancer
80 of every 100 women with breast cancer will
get positive tests
950 out of 9,900 women without breast cancer
will also get positive tests
PROBLEM
If 10,000 women in this age group undergo a
routine screening, about what fraction of
women with positive tests will actually have
breast cancer?
Bayesian Reasoning
Before the screening:
100 out of 10000women with breast cancer
9,900 out of 10000women without breast cancer
After the screening:
A = 80 out of 10000women with breast cancer and
positive test
B = 20 out of 10000 women with breast cancer and
negative test
C = 950 out of 10000 women without breast cancer
and positive test
D = 8,950 out of 10000women without breast cancer
and negative test
All patients that has positive test result = A+C
Proportion of cancer patients with positive results,
within the group of ALL patients with positive results:
A/(A+C) = 80/(80+950) = 80/1030 = 0.078 = 7.8%
Bayesian Reasoning
Prior Probabilities:
100/10,000 = 1/100 = 1% = p(A)
9,900/10,000 = 99/100 = 99% = p(~A)
Conditional Probabilities:
A = 80/10,000 = (80/100)*(1/100) = p(B|A)*p(A) = 0.008
B = 20/10,000 = (20/100)*(1/100) = p(~B|A)*p(A) = 0.002
C = 950/10,000 = (9.6/100)*(99/100) = p(B|~A)*p(~A) =
0.095
D = 8,950/10,000 = (90.4/100)*(99/100) = p(~B|~A) *p(~A) =
0.895
Rate of cancer patients with positive results, within
the group of ALL patients with positive results:
P(A|B) = P(B|A) * P(A)/ P(B) = 0.008/(0.008+0.095) =
0.008/0.103 = 0.078 = 7.8%
Another example
• Suppose that we are interested in diagnosing
cancer in patients who visit a chest clinic:
• Let A represent the event "Person has cancer"
• Let B represent the event "Person is a smoker"
• We know the probability of the prior event
P(A)=0.1 on the basis of past data (10% of
patients entering the clinic turn out to have
cancer). We want to compute the probability of
the posterior event P(A|B). It is difficult to find
this out directly. However, we are likely to know
P(B) by considering the percentage of patients
who smoke – suppose P(B)=0.5. We are also likely
to know P(B|A) by checking from our record the
proportion of smokers among those diagnosed.
Suppose P(B|A)=0.8.
• We can now use Bayes' rule to compute:
• P(A|B) = (0.8 * 0.1)/0.5 = 0.16
• Thus we found the proportion of cancer patients
who are smokers.
Bayes in Brain Imaging
Extension to distributions:
likelihood prior
𝑃 𝐵|𝐴 ∙ 𝑃 𝐴
𝑃 𝐴|𝐵 =
𝑃 𝐵
posterior
marginal
probability
Bayes in Brain Imaging
Extension to distributions:
likelihood
prior distribution
𝑃 𝐵|𝐴
𝑃 𝑑𝑎𝑡𝑎|𝜃
∙𝑃 𝐴 ∙𝑃 𝜃
𝑃 𝜃|𝑑𝑎𝑡𝑎
𝐴|𝐵 = =
𝑃 𝐵𝑃 𝑑𝑎𝑡𝑎
posterior distribution
𝜃
marginal probability
is a set of parameters defining a model
Bayes in Brain Imaging
likelihood
𝑃 𝜃|𝑑𝑎𝑡𝑎 =
prior distribution
𝑃 𝑑𝑎𝑡𝑎|𝜃 ∙ 𝑃 𝜃
𝑃 𝑑𝑎𝑡𝑎
posterior distribution
Example: How many infections should a hospital
expect over 40,000 bed-days?
Data from other hospitals show 5 to 17
infections per 10,000 bed-days:
Prior distribution
Over the first 6 months of the year, 4
infections per 10,000 bed-days:
Likelihood distribution
*
Product of the distributions:
Posterior distribution
Answer is calculated using the posterior distribution
*Spiegelhalter
and bed-days
Rice
Number
of infections for 40,000
Bayes in Brain Imaging
Classical Approach:
Bayesian Approach:
• Null hypothesis is that no
activation occurred
• Use the likelihood and prior
distributions to construct the
posterior distribution
• Statistics performed, e.g. a Tstatistic
• Reject null hypothesis if data is
sufficiently unlikely
This approach gives the likelihood of
getting the data, given no
activation.
Cannot accept the null hypothesis that
no activation has occurred.
• Compute probability that
activation exceeds some
threshold directly from the
posterior distribution.
This approach gives the probability
distribution of the activation given the
data.
Can determine activation / no activation.
Can compare models of the data.
Bayes Example 1
The GLM: Estimating Beta’s
1
p
1
1
β
y
N
Observed Signal/Data =
=
N
X
p
+
ε
N
Experimental Matrix x Parameter Estimates(prior) + Error (Artifact, Random Noise)
Bayes Example 1
The GLM: Estimating Beta’s
Bayes Example 1
The GLM: Estimating Beta’s
Bayes Example 2
Example 2: Accepting/Rejecting Null Hypothesis
Activated
Not Activated
*Dr. Joerg Magerkurth
Bayes Example 3
Segmentation and Normalization
• Tissue probability maps of gray and white matter used
as priors in the segmentation algorithm.
• Probability of expected zooms and shears used as prior
in normalization algorithm.
Thanks to Will Penny and Previous
MfD Presentations
Questions?